### Abstract

Let Σ be a (connected) surface of "complexity" κ; that is, Σ may be obtained from a sphere by adding either 1/2κ handles or κ crosscaps. Let ρ ≥ 0 be an integer, and let Γ be a "ρ-representative drawing" in Σ; that is, a drawing of a graph in Σ so that every simple closed curve in Σ that meets the drawing in <ρ points bounds a disc in Σ. Now let Γ′ be another drawing, in another surface Σ′ of complexity κ′, so that Γ and Γ′ are isomorphic as abstract graphs. We prove that (i) If ρ ≥ 100 log κ/ log log κ (or ρ ≥ 100 if κ ≤ 2) then κ′ ≤ κ, and if κ′ = κ and Γ is simple and 3-connected there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, (ii) if Γ is simple and 3-connected and Γ′ is 3-representative, and ρ ≥ (320, 5 log κ), then either there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, or κ′ ≥ κ + 10^{-4} ρ^{2}.

Original language | English (US) |
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Pages (from-to) | 337-349 |

Number of pages | 13 |

Journal | Journal of Graph Theory |

Volume | 23 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1996 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

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## Cite this

*Journal of Graph Theory*,

*23*(4), 337-349. https://doi.org/10.1002/(SICI)1097-0118(199612)23:4<337::AID-JGT2>3.0.CO;2-S